This site is devoted to mathematics and its applications. Created and run by Peter Saveliev.

# New vector spaces from old

### From Mathematics Is A Science

The main examples are the following:

Given a vector space X and a subset Y of X. Then Y is called a subspace of X if it's a vector space. The inclusion i_{Y}: Y → X, i_{Y}(x) = x, is a linear operator.

Given two vector spaces X and Y. Then X×Y is the product of sets, set of all pairs (a,b) of elements in X and Y respectively. It is a vector space with the operations (a,b) + (a',b') = (a + a',b + b') and t(a,b) = (ta,tb). The projections p_{X}: X×Y → X and p_{Y}: X×Y → Y, p_{X}(a,b) = a, p_{Y}(a,b) = b, are linear operators.

Given a vector space X and a subspace Y. Then an equivalence relation ~ on X is defined by x ~ y if x - y ∈ Y. Then the quotient set X/~ is a vector space with the operation [x] + [y] = [x + y] and q[x] = [qx]. The quotient function q: X → X/~, q(x) = [x], is a linear operator.

See also: